Parabolic equations in Musielak - Orlicz spaces with discontinuous in time N-function

We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ non-standard growth controlled by an $N$-function depending on time spatial variable. do not assume continuity in for the $N$-function. Using additional regularization effect coming from equation, we establish existence of weak solutions particular case isotropic $N$-function, also prove their uniqueness. This general result applies to equations studied literature like $p(t,x)$-Laplacian double-phase problems.